Interior regularity results (local) and Template:Ombox/core: Difference between pages

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'''The math formulas do not work!!!'''
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Let <math>\Omega</math> be an open domain and <math> u </math> a solution of an elliptic equation in <math> \Omega </math>. The following theorems say that <math> u </math> satisfies some regularity estimates in the interior of <math> \Omega </math> (but not necessarily up the the boundary).
{{documentation}}
 
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* [[De Giorgi - Nash - Moser]]
</noinclude>
<math> div A Du + b Du = 0 </math>
then <math>u</math> is Holder continuous if <math>A</math> is just uniformly elliptic and <math>b</math>
is in <math>L^n</math> (or <math>BMO^{-1}</math> if <math>div b=0</math>).
 
* [[Krylov-Safonov]]
<math>a_ij(x) u_ij + b Du = f</math>
with <math>a_{ij}</math> unif elliptic (L^infty), b in L^n and f in L^n, then the
solution is C^alpha
 
* [[Calderon-zygmund]]
a_ij(x) u_ij = f
with a_ij close enough to the identity and f in L^p, then w is in W^{2,p}.
I am not sure how lower order terms affect the result. It should be
somewhere in [GT]
 
* [[Cordes-Nirenberg]]
a_ij(x) u_ij = f
with a_ij close enough to the identity uniformly and f in L^infty,
then w is in C^{1,alpha}
I am pretty sure you can add first order terms as long as the
coefficients are small enough in L^infyt
 
* [[Cordes-Nirenberg improved]] (corollary of work of Caffarelli for nonlinear equations)
<math>a_ij(x) u_ij = f</math>
with a_ij close enough to the identity in a scale invariant Morrey
norm in terms of L^n and f in L^n, then w is in C^{1,alpha}
(VMO is a particular case of this)
 
*[[Schauder]]
<math>a_ij(x) u_ij = f</math>
with a_ij in C^alpha and f in C^alpha, then u is in C^{2,alpha}
I don't remember the hypothesis for the first order terms, but this is
in [GT] for sure.

Latest revision as of 17:47, 20 May 2011

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